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  "body": "\n## Summary\n\nThe Zariski tangent space of a variety at a point is the linear space of directions in which the defining equations vanish to first order.\nFor a variety in $k^n$ it is the kernel of the Jacobian matrix of the defining polynomials at the point.\nIntrinsically it is the dual of $\\mathfrak m / \\mathfrak m^2$, where $\\mathfrak m$ is the maximal ideal of the point.\nIts dimension is at least the dimension of the variety, with equality exactly at the nonsingular points.\n\n## Explanations\n\n### L1 human\n\nNear a point, replace each defining equation by its best linear approximation, its differential at the point.\nThe common solutions of these linear equations form the tangent space.\nFor the circle $x^2 + y^2 = 1$ at $(1, 0)$, the differential is $2x \\, dx + 2y \\, dy = 2 \\, dx$, so the tangent space is the vertical line $dx = 0$.\nFor the cusp $y^2 = x^3$ at the origin, all the differentials vanish, so the tangent space is the whole plane, larger than the curve's dimension $1$.\n",
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